The emergence of graphene, a nanomaterial with a two-dimensional structure, ignited enthusiasm for research in the realm of two-dimensional materials. It is the world's first two-dimensional material, isolated from graphite in 2004 by Professors Andre Geim and Kostya Novoselov [1]. It serves as the fundamental building block for various other carbon-based structures, such as graphite, charcoal, carbon nanotubes, and fullerenes. Additionally, it can be viewed as a huge aromatic molecule, representing the ultimate form of a class of planar polycyclic aromatic hydrocarbons. The discovery of graphene has significantly impacted the field of materials, greatly enhancing the utilization of two-dimensional materials across diverse fields and applications. Remarkably, graphene is 200 times stronger than steel and incredibly thin–one million times thinner than human hair. At present, numerous products available in the market leverage the prominent influence of graphene, which often plays a substantial or even primary role in enhancing the properties and functionalities of these products. For instance, graphene and other two-dimensional materials can be utilized to create microwave shields and absorbers with specific, predetermined properties [2]. Its unrivaled conductivity has sparked global interest among scientists, researchers, and industries. With its extraordinary properties, graphene holds immense promise across diverse fields, from electronics and optics to sensors and biomedical devices. Notably, it stands out as the top-performing material for shielding against electromagnetic interference (EMI) (see [3–6] for further details). Graphenes have been a subject of discussion in chemical graph theory over the years. The establishment of the fundamental principles of contemporary chemical graph theory (CGT) can be attributed to the groundbreaking contributions of mathematicians Arthur Cayley (1821–1895) and James Sylvester (1814–1897). In their effort to describe chemical structures using purely mathematical concepts, they determined that an isomorphism exists between chemical structures and mathematical graphs [7]. With the exception of certain mathematical disciplines, chemistry represents the primary domain where graph theory finds extensive application. The immense importance of graph theory in the field of chemistry has gained widespread recognition, particularly in the realm of CGT, where considerable attention is devoted to renowned topological indices (TIs) or index (TI). TIs are typically categorized into distance-based, entropy-based, spectral-based, and degree-based indices. TIs find extensive application in correlating and predicting a wide array of molecular properties through QSPR analysis (see [8–14] for further details). Harold Wiener introduced the initial distance-based TI known as the Wiener index in 1947. Wiener's index gained prominence when his experiment revealed a significant correlation between the boiling points of alkane molecules and the corresponding Wiener index values [9]. Since then, a multitude of degree-based TIs have been developed in the field of CGT, with numerous publications discussing these advancements with aid to create more TIs that accurately correlate with the structural properties of chemical compounds (see [12, 15–28] for more details).

In a graph ℌ = (V, E), with V(ℌ) representing the vertices (such as carbon atoms) and E(ℌ) representing the edges (such as bonds), the degree d_u of a vertex u ∈ V(ℌ) is the count of edges incident to it. The neighborhood of a vertex u, denoted by N(ℌ)_u, refers to the set containing all vertices v that are directly connected to u. The neighborhood sum degree of a vertex u is defined as the sum of the degrees of the neighboring vertices connected to u, expressed as:

In recent years, researchers have increasingly focused on designing TIs that are based on the neighborhood degree of vertices. In 2019, Mondal et al. [29] introduced the first class of neighborhood versions of degree-based TIs, including the neighborhood second Zagreb index, the neighborhood forgotten index, the modified neighborhood forgotten index, and the neighborhood hyper-Zagreb index. For example, let ℌ be a graph. The first and second neighborhood Zagreb indices of ℌ are defined as follows:

In their QSPR analysis, Mondal et al. [29] observed that the newly introduced neighborhood TIs outperformed their degree-based TIs counterparts when applied to model the physicochemical properties of octane isomers. Subsequently, Mondal et al. [30] additionally proposed a range of neighborhood versions of the following degree-based TIs, specifically the second modified Zagreb index, the general Randić index, the harmonic index, and the inverse sum index, which were derived from the neighborhood M-polynomials of crystallographic structures. Further research into the extension of neighborhood degree-based TIs can be found in the following works [31–35].

TIs can be computed through direct computation or by using algebraic polynomials. However, employing direct computation to compute TIs can become cumbersome when deriving multiple TIs within a specific category. To address this issue, numerous algebraic polynomials [36–40] have been introduced, from which TIs can be derived by differentiation, integration, or a combination of both at an invariant point. Chemistry also benefits from the valuable applications of algebraic polynomials, such as the Hosoya polynomial (commonly referred to as the Wiener polynomial). For instance, evaluating the derivatives of the Hosoya polynomial at x = 1 yields both the Wiener index and the Hyper Wiener index [37]. This polynomial is considered instrumental in determining distance-based TIs. Among various algebraic polynomials, Deutsch and Klavžar [41] introduced the concept of the M-polynomial in 2015. The M-polynomial is the most versatile polynomial for generating a wide range of degree-based TIs, providing closed-form expressions for degree-based TIs. In 2017, Kwun et al. [42] derived degree-based TIs using the M-polynomial method for V-phenylenic nanotubes and nanotori. Since then, the versatility of the M-polynomial has attracted significant research attention, leading to the derivation of numerous TIs for diverse graph structures. These include line graphs of subdivision graphs [43], linear chains of aromatic hydrocarbons like benzene, naphthalene, and anthracene [44], antituberculosis drugs [45], and borophene nanosheets [14], among others. Every day, significant progress is made in the development of new TIs. One notable advancement is the concept of the neighborhood M-polynomial (NM), which emerges from the field of neighborhood degree-based TIs. This concept was initially introduced by Mondal et al. in their 2019 works [30]. The NM- polynomial stands out from the classical M-polynomial by employing a neighborhood edge partitioning system to divide the graph's edges. The NM-polynomial defined by Mondal et al. [30] for the graph ℌ is given by:

where r_ik(ℌ), i, k ≥ 1 denotes the total number of edges e = uv of ℌ such that {λ_u, λ_v} = {i, k}.

Several studies have explored the application of NM-polynomials to obtain neighborhood TIs for various molecular structures. Verma et al. [46] employed the NM-polynomial to investigate the topological properties of bismuth tri-iodide. In 2020, Mondal et al. [47] obtained TIs for chemical structures for the treatment of COVID-19 via M-polynomial and NM-polynomial. Shanmukha et al. [48] determined the M-polynomial and NM-polynomial to derive TIs of porous graphene. Molecular descriptors based on the neighborhood degree sum for fractal and Cayley tree dendrimers were obtained by Mondal et al. [49] via NM-polynomial. Present authors [50] obtained neighborhood TIs using NM-polynomial of graphene networks and their line graph of graphene networks. A recent study by Xavier et al. [51] conducted a comparative analysis of pent-heptagonal nanostructures using TIs derived from the NM-polynomial approach. Most recently in 2024, Abubakar et al. [38] explored the applications of NM-polynomials in characterizing oligothiophene dendrimers. This expanding area of research highlights the promise of NM-polynomials in characterizing different molecular structures. However, a thorough understanding of neighborhood TIs is still being developed, with ongoing efforts to enhance their efficiency. Addressing this gap, this paper introduces the neighborhood multiple M-polynomial, an extension of the NM-polynomials, which will be applied to characterize the structure of graphene. This approach will retrieve eleven neighborhood versions of multiple degree-based TIs listed in Table 1, utilizing neighborhood multiple-edge partition sets.

Let ℌ = (V, E) be a graph. The following definitions are adaptations of the multiple degree TIs proposed by Gao et al. [52].

Definition 2.1. The neighborhood multiple degree function ξ_u = ∏_v∈N(ℌ)u d_v is defined as the product of the degrees of all vertices v in the neighborhood of vertex u.

Definition 2.2. In a graph ℌ, the neighborhood multiple-edge partition sets for all edges uv ∈ E(ℌ) are defined by the pairs (ξ_u, ξ_v), where ξ_u and ξ_v are the neighborhood multiple degree functions of the vertices u and v, respectively.

The following definitions are a reformulation of neighborhood M-polynomial and neighborhood sum degree-based TIs by Mondal et al. [30].

Definition 2.3. Let ℌ be a graph, the neighborhood multiple M-polynomial (N_MM) is defined as,

where τ_ik(ℌ), i, k ≥ 1 denotes the total number of edges e = uv of ℌ_v such that {ξ_u, ξ_v} = {i, k}, and p, q denotes the variables.

Neighborhood multiple degree-based TIs defined by using neighborhood multiple degree edge partition in an arbitrary graph ℌ can be defined as,

where f(ξ_u, ξ_v) represents the function of ξ_u and ξ_v employed in definition of neighborhood multiple degree-based TIs in Definition 2.4. An alternative way to represent the previous result is,

Now, let's provide an example to illustrate the previously mentioned definitions.

Example 2.1. Consider the undirected graph ℌ from Figure 1, such that |V(ℌ)| = 6 and |E(ℌ)| = 7.

By Definition 2.1, we have ξ₁ = 9, ξ₂ = 12, ξ₃ = 9, ξ₄ = 6, ξ₅ = 18, and ξ₆ = 3. Thus, by Definition 2.2, we partition the edge sets via the neighborhood multiple degrees of the end vertices for all τ_{(ξu, ξv)}: τ_(9,12) = 2, τ_(6,9) = 1, τ_(6,18) = 1, τ_(9,18) = 1, and τ_(3,6) = 1. Therefore, from Equation 2.1, we obtain

Definition 2.4. Let ℌ be a graph.

Table 1 displays the relationship between neighborhood multiple TIs and neighborhood multiple M-polynomial using the function defined in Definition 2.3, with α=12.

Where,

This study focuses on the neighborhood multiple degree-based TIs of the graphene structure when t > 1 and s > 1. First, we calculate the neighborhood multiple M-polynomial. By leveraging integration and differentiation techniques, we then derive eleven neighborhood multiple-degree-based TIs. This is achieved using graph theory algorithms, combinatorial computation, and neighborhood multiple-degree partition sets to extract the results. We employed Matlab 24.1 to calculate all neighborhood multiple degree-based TIs values for various graphene structures (t, s). The resulting TIs are presented with comprehensive analysis through graphical and tabular comparisons. Specifically, 3D plots of the neighborhood multiple M-polynomial and the visualization of numerical values were created using Matlab 24.1, while comparison scatter plots were generated using Python 3.12.5.

Graphene's structure resembles a hexagonal lattice as shown in Figure 2. In this lattice, the vertices correspond to carbon atoms, and the edges represent their covalent bonds. This honeycomb-like arrangement consists of t rows and s columns, forming a total of ts hexagons. Mathematically, the entire structure can be represented as ℌ(t, s). The structure of graphene consists of four cases:

Case 1: t > 1, s > 1, Case 2: t = 1, s = 1, Case 3: t = 1, s > 1, and Case 4: t > 1, s = 1. In each case, the values of t and s determine the specific structure of graphene.

However, this study investigates only the instance of Case 1 of the graphene structure. To achieve this, we sort the neighborhood multiple edge partition sets τ_{(ξu, ξv)} from the graphene structure shown in Figure 2, resulting in Table 2.

Theorem 4.1. Let ℌ be a graphene structure (t, s). The neighborhood multiple M-polynomial is given by:

Proof. By examining the graphene structure (t, s) reveals that the corresponding graph ℌ, possesses a total of 2ts + 2t + 2s vertices and 3ts + 2t + 2s − 1 edges. Furthermore, by Definition 2.1 the neighborhood multiple degrees of vertices within ℌ can take on the following distinct values: 4, 6, 9, 12, 18, and 27. This observation leads to the identification of nine distinct neighborhood multiple-edge partition sets for all edges uv ∈ E(ℌ) defined by the pairs (ξ_u, ξ_v), respectively. The identified partition sets are τ_(4,6), τ_(6,6), τ_(6,12), τ_(6,18), τ_(9,12), τ_(12,27), τ_(18,18), τ_(18,27), and τ_(27,27) as shown in Figure 2. Table 2 details these neighborhood multiple-edge partition sets and the corresponding number of edges within each partition.

The neighborhood multiple M-polynomial of ℌ, derived from Definition 2.3 and utilizing Table 2, is obtained as follows:

Thus, the proof is complete.

Figure 3 visually represents the aforementioned result, with the given parameters t=6 and s=9 of graphene structure. The plot visualizes how the function behaves over the range of p and q from 10 to -10.

Remark 4.1. For simplicity, we denote N_MM(ℌ; p, q) as γ(p, q) throughout this work.

Theorem 4.2. Let ℌ be a graphene structure when t > 1 and s > 1, then we have:

Proof. The neighborhood multiple M-polynomial of ℌ when t > 1 and s > 1 is given as follows,

Let γ(p, q) = N_MM(ℌ; p, q). Thus, we have:

Therefore, from Table 1, we get:

Theorem 4.3. Consider the graphene structure for t > 1 and s > 1, then the following results hold:

Proof. The proof follows a similar approach to Theorem 4.2. Define γ(p, q) = N_MM(ℌ; p, q). Therefore, we obtain:

Theorem 4.4. Let ℌ be a graphene structure when t > 1 and s > 1, then we have:

Proof. Define γ(p, q) = N_MM(ℌ; p, q). Using Table 1, we obtain:

The veil of uncertainty surrounding the properties of 2D nanomaterials like graphene is thinning thanks to the power of molecular descriptors. These computational tools act as crystal balls, allowing scientists to peer into a material's potential based on its atomic arrangement. Particularly in graphene research, molecular descriptors play a crucial role in property prediction. While these descriptors can incorporate aspects of 3D electronic structure (relevant for some properties), their primary strength lies in capturing the unique features of graphene's 2D structure. Different categories of descriptors, ranging from simple measures of connectivity (1D) to complex electronic considerations, offer valuable insights into various aspects of graphene's behavior, such as its exceptional electrical conductivity, thermal stability, or potential for gas adsorption. Additionally, advanced methods can generate quantum-chemical descriptors. These descriptors delve into the electronic structure of the molecule, providing even deeper insights (refer [53, 54]). The recent surge in artificial intelligence and machine learning has opened doors for developing even more sophisticated descriptors and identifying the most informative ones for specific applications (refer [55, 56]). The core principle of QSPR is that a molecule's structure holds the key to its properties. QSPR methods aim to capture these properties using mathematical codes called descriptors. QSPR modeling rely on a vast library of pre-calculated descriptors, encompassing both physical and structural properties of molecules. This computational approach is much faster and more cost-effective than traditional experimental methods, which often require significant effort and expensive equipment. QSPR analysis workflow is depicted as a flowchart in Figure 4. Notably, the workflow emphasizes the utilization of structural descriptors specifically tailored for graphene structure.

Building upon the established interest in graphene structures, this section focuses on developing linear regression models to predict various properties of these materials. Here, we employ the least squares method to establish relationships between a selection of properties and corresponding neighborhood multiple degree-based TIs of the graphene structures. The following equation serves as the foundation for our analysis:

where Z is a mechanical property of the nanomaterial, n is the regression coefficient, m is a constant, and TI is a neighborhood multiple degree-based TI. Computational approaches, such as theoretical analysis, have the potential to aid pharmaceutical scientists, including chemists, in the prediction of nanomaterial features, thereby reducing the need for extensive experimentation. This also facilitates the design of new nanomaterials with desired properties. The neighborhood multiple M-polynomial technique offers a more efficient and concise approach to TIs computation for graphene structures compared to traditional algorithmic methods. The efficiency of the neighborhood multiple M-polynomial approach derives from its unique methodology. Unlike traditional methods that require separate calculations for each neighborhood multiple TI, this technique enables the derivation of numerous TIs through differentiation and integration, eliminating the need for repetitive computations.

The Least Squares Approximation method was employed to elucidate the relationship between neighborhood multiple degree-based TIs and the properties of various graphene variations, as detailed in Table 5. Correlation, a statistical measure, quantifies the strength and direction of a linear relationship between two variables via the correlation coefficient, r, which ranges from -1 to 1. An r value of 1 signifies a strong positive linear relationship, -1 indicates a strong negative linear relationship, and 0 denotes no linear relationship. Regression analyses were performed using analytical tools such as Matlab 24.1 and Python 3.12.5. The resulting correlation coefficients, r, are summarized and visualized in Tables 6, 7, Figure 6. See Figure 7 also.